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Quantum theory of plasmon

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2014 Adv. Nat. Sci: Nanosci. Nanotechnol. 5 025001

(http://iopscience.iop.org/2043-6262/5/2/025001)

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Quantum theory of plasmon

Van Hieu Nguyen and Bich Ha Nguyen

Institute of Materials Science, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, CauGiay, Hanoi, Vietnam

E-mail: nvhieu@iop.vast.ac.vn

Received 10 February 2014Accepted for publication 4 March 2014Published 26 March 2014

AbstractSince very early works on plasma oscillations in solids, it was known that in collectiveexcitations (fluctuations of the charge density) of the electron gas there exists the resonanceappearing as a quasiparticle of a special type called the plasmon. The elaboration of the quantumtheory of plasmon in the framework of the canonical formalism is the purpose of the presentwork. We start from the establishment of the Lagrangian of the system of itinerant electrons inmetal and the definition of the generalized coordinates and velocities of this system. Then wedetermine the expression of the Hamiltonian and perform the quantization procedure in thecanonical formalism. By means of this rigorous method we can derive the expressions of theHamiltonians of the interactions of plasmon with photon and all quasiparticles in solid from thefirst principles.

Keywords: plasmon, canonical formalism, fluctuation, collective excitationClassification numbers: 2.09, 3.00, 3.02

1. Introduction

The existence of the resonance in collective excitations of theelectron gas in metals was demonstrated in early works onplasma oscillations in solids (see, for example, [1–4]). Itbehaves like a quasiparticle of a special type called plasmon.Recently the research on the fundamental processes with theparticipation of plasmons has led to the emergence of a newscientific discipline —plasmonics. The variety of plasmonicprocesses and phenomena is quite broad: the formation ofhybrid systems consisting of semiconductor quantum dot andmetallic nanoparticles [5, 6], interaction between a metalnanoparticle and a dipole emitter [7], exiton–plasmon cou-pling (plexciton) [8–10], plasmon resonance energy transfer(PRET) [11, 12], plasmon-enhanced light absorption [13, 14]and fluorescene [15–20], plasmonic-molecular resonance[21–28] etc. The results of the research on plasmonic pro-cesses have led to the creation of plasmonic nanoantenae forvarious efficient applications [29].

In many of the above-mentioned fundamental researchworks on plasmonic processes there was the need to use theHamiltonians of the interactions between plasmon and otherelementary excitations in matter. All those interaction

Hamiltonians were introduced in a phenomenological man-ner. In order to exactly derive the Hamiltonians of the inter-actions of plasmon from first principles it is necessary toelaborate a rigorous procedure for quantizing the collectiveexcitations of electron gas. This task will be performed in thepresent work. The main ideas were outlined in our previouspublication [30].

2. Lagrangian of the system of itinerant electrons inmetal

Consider a simple model of metal consisting of a gas ofitinerant electrons freely moving inside metal over a back-ground of ions with homogeneous distribution of the positivecharge. Denote n tr( , ) the electron density (number ofelectrons per unit volume) and n0 its mean value (averagedover space and time). The average charge density −en0 ofelectron, −e being the electron charge, compensates theaverage positive charge of the ions in the background, and the

2043-6262/14/025001+06$33.00 © 2014 Vietnam Academy of Science & Technology1

| Vietnam Academy of Science and Technology Advances in Natural Sciences: Nanoscience and Nanotechnology

Adv. Nat. Sci.: Nanosci. Nanotechnol. 5 (2014) 025001 (6pp) doi:10.1088/2043-6262/5/2/025001

fluctuating charge density in the metal is

ρ = − −t e n t nr r( , ) [ ( , ) ] . (1)0

According to the Coulomb law the effective charge dis-tribution (1) creates a time-dependent electrical field with thepotential

∫φρ

= ′′

− ′⋅t d

tr r

r

r r( , )

( , )(2)

From formula (2) there follows the Poisson equation

φ πρ= − t tr r( , ) 4 ( , ) . (3)2

The mutual interaction between effective charge densitiesat two different regions in the space gives rise to the potentialenergy of the electron gas

∫ ∫ ρ ρ= ′− ′

′U t d d t tr r rr r

r( )1

2( , )

1( , ) , (4)

which can be also written in the form

∫ ρ φ=U t d t tr r r( )1

2( , ) ( , ) . (5)

As a consequence of the oscillating displacements ofelectrons, the fluctuation of the electron density n tr( , )generates the total kinetic energy of the electron gas. Denoteδ tr r( , ) the displacement of the electron having the coordi-nate r at the time moment t, and m the electron mass. Sincethe electron has the velocity

δδ =

∂∂

tt

tr r

r r( , )

( , ), (6)

the whole electron gas has following total kinetic energy

∫ δ =T t m d n t tr r r r( )1

2( , ) ( , ) . (7)2

Thus we have derived following expression of theLagrangian of the electron gas

∫∫ ∫

δ

ρ ρ

=

− ′− ′

′

L t m d n t t

d d t t

r r r r

r r rr r

r

( )1

2( , ) ( , )

1

2( , )

1( , ) , (8)

2

where the spatial integrations are performed over the wholevolume of the electron gas. Replacing the expression

ρ= −n t ne

tr r( , )1

( , )0

into the r.h.s. of formula (8), we rewrite this formula in theform containing only two types of dynamical variables—theelectron displacement vectors δ tr r( , ) and the charge density

ρ tr( , ) :

∫∫ ∫

∫

δ

ρ ρ

ρ δ

=

− ′− ′

′

−

L t mn d t

d d t t

m

ed t t

r r r

r r rr r

r

r r r r

( )1

2( , )

1

2( , )

1( , )

1

2( , ) ( , ) .

. (9)

02

2

The dynamical variables δ tr r( , ) and ρ tr( , ) cannot becompletely independent, and we must establish the relation-ship between these physical quantities. We note that sinceoscillating displacements of electrons cause the fluctuations ofthe electron number n tr( , ) , there must exist some directrelationship between n tr( , ) and δ tr r( , ) . For establishingthis relationship we consider any finite volume Ω bounded bya closed surface Σ in the spatial region of the electron gas. Incomparison with the average electron number density n0needed for compensating the positive charge of the ions in thelattice of the metal, the number of excess electrons in thevolume Ω is

∫= −ΩΩ

N t d n t nr r( ) [ ( , ) ] . (10)0

Due to the fluctuation of n tr( , ) , this number changeswith the time and its increment during a very short timeinterval (t, t+ δt) is

∫δ δ= ΩΩ

N t d n t tr r( ) ( , ) , (11)

=∂

∂⋅n t

n t

tr

r( , )

( , )(12)

On the other side, the fluctuation of n tr( , ) is caused bythe oscillating displacements of electrons. Denote δ ΣN t( ) thenumber of electrons displacing across the boundary Σ of thevolume Ω and leaving this volume, i.e. moving from theinside of the closed surface Σ to its outside during the sametime interval (t, t + δt). We have

∮δ δ δ=ΣΩ

N t d t n t tS r r r( ) ( , ) ( , ) . (13)

By means of the Ostrogradski–Gauss theorem we trans-form the surface integral over Σ in the r.h.s. of formula (13)into a volume integral over Ω and obtain

∫δ δ δ=ΣΩ

N t d n t t tr r r r( ) [ ( , ) ( , ) ] . (14)

Because the number δ ΣN t( ) of electrons leaving thevolume Ω across its surface Σ must be equal to the decrement

δ− ΩN t( ) of the number of electrons contained in the volumeΩ

δ δ= −Σ ΩN t N t( ) ( ) ,

Adv. Nat. Sci.: Nanosci. Nanotechnol. 5 (2014) 025001 V H Nguyen and B H Nguyen

2

we obtain following formula

δ

=∂

∂= −

n tn t

tn t t

rr

r r r

( , )( , )

[ ( , ) ( , )] (15)

called the continuity equation. In terms of the charge density(1), the continuity equation has the form

ρρ

δρ δ

=∂

∂=

−

tt

ten t

t t

rr

r rr r r

( , )( , )

( , )

[ ( , ) ( , ) ]. (16)0

Thus the dynamical variables δ tr r( , ) and ρ tr( , ) in theLagrangian (9) are not independent. They must satisfy thesubsidiary condition (16).

The appearance of the small effective charge densityρ tr( , ) in comparison with the average charge density(having the absolute value en0) of the electron gas is theconsequence of the very small oscillating displacementsδ tr r( , ) of the electrons. In the Lagrangian (9) and the sub-sidiary condition (16) they are two very small quantities of thesame order. Consider Lagrangian (9) and the subsidiarycondition (16) in the lowest order with respect to these verysmall quantities. Then the Lagrangian has the followingapproximate expression

∫∫ ∫

δ

ρ ρ

′

=

−− ′

′

L t mn d t

d d t t

r r r

r r rr r

r

( )1

2( , )

1

2( , )

1( , ) , (17)

0 02

and the subsidiary condition becomes

ρ δ = t en tr r r( , ) ( , ) . (18)0

3. Canonical formalism in the harmonicapproximation

Now we establish the canonical formalism of classicalmechanics and then apply the quantization procedure to thestudy of the electron gas with the charge density ρ tr( , ) andthe electron displacement vectors δ tr r( , ) being the solutionof the system of differential equations consisting of Lagrangeequations with the approximate Lagrangian (17) and thesubsidiary condition (18). For this purpose we decomposefunctions ρ tr( , ) and δ tr r( , ) into the Fourier series of planewaves normalized in a cube with the volume V and ortho-gonalized by means of the periodic boundary conditions:

∑ρ ρ=tV

t er( , )1

( ) , (19)i

kk

kr

∑ ∑δ =α

α α

=

tV

f t er r e( , )1

( ) (20)i

kk k

kr

1

3( ) ( )

where αek( ) for each pair of indices k and α, α = 1, 2, 3, are

three real unit vectors satisfying the orthogonalization andnormalization conditions

δ=α ααβe e (21)k k

( ) ( )

and having properties

=

=k

ke

ek0,

,(22)

( )

( )

k

k

1,2

3

= −α α−e e . (23)k k( ) ( )

Since ρ tr( , ) and δ tr r( , ) are the real quantities, theirFourier transforms must satisfy conditions

ρ ρ= *

= *α α−

−

t t

f t f t

( ) ( ) ,

( ) ( ) . (24)k k

k k( ) ( )

In terms of the Fourier components ρ t( )k

and αf t( )k( ) the

subsidiary condition (18) becomes

ρρ

= = td t

dtien k f t( )

( )( ) . (25)( )

kk

k0

3

Substituting the Fourier series (19) and (20) into the r.h.s.of formula (17), we obtain

⎡⎣⎢

⎤⎦⎥

∑ ∑

π ρ ρ

= *

− *

α

α α

=

L t mn f t f t

kt t

( )1

2( ) ( )

4( ) ( ) . (26)

kk k

k k

0 0

1

3( ) ( )

2

Using formula (25) to express f t( )( )k

3in the r.h.s. of

relation (26) in terms of ρ t( )k

, we rewrite the Lagrangian

L0(t) in the new form containing f t( )( )k

1,2, ρ t( )

kand ρ t( )

k:

⎡⎣⎤⎦

∑

∑

ρ ρ

ω ρ ρ

=

× *

+ *

− *

α

α α

=

}

L t mn

f t f t

e

m

n kt t

t t

( )1

2{

( ) ( )

1 1( ) ( )

( ) ( ) (27)pl

k

k k

k k

k k

0 0

1,2

( ) ( )

20

2

2

with

ω π= e m

n

4(28)pl

2

0

called the plasma frequency.From the Lagrangian equations

⎛

⎝⎜⎜

⎞

⎠⎟⎟

∂

∂ =

∂∂

d

dt

L t

f t

L t

f t

( )

( )

( )

( ),

( ) ( )k k

0

1,2

0

1,2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

∂

∂ *=

∂∂ *

d

dt

L t

f t

L t

f t

( )

( )

( )

( )( ) ( )k k

0

1,2

0

1,2

and expression (27) of L0(t) it follows that the generalized

velocities f ( )k

1,2and *

f ( )k

1,2must be time-independent. Because

Adv. Nat. Sci.: Nanosci. Nanotechnol. 5 (2014) 025001 V H Nguyen and B H Nguyen

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the addition of arbitrary constants to the Lagrangian L0(t)does not affect the equation of motion of the system, the terms

containing the constants f ( )k

1,2and *

f ( )k

1,2in the r.h.s. of for-

mula (27) can be discarded. The absence of these constants inthe Lagrangian has the following clear physical meaning:only the longitudinal displacements of electrons can cause thewave of propagating fluctuations of electron number densityn tr( , ) , i.e. of charge density, in the electron gas. Thereforeinstead of the expansion formula (20) we shall use followingexpression

∑δ =tV k

f t er rk

( , )1

( ) . (29)i

kk

kr

In this case the Lagrangian (26) becomes

⎡⎣⎤⎦⎥

∑π ρ ρ

= *

− *

L t mn f t f t

kt t

( )1

2( ) ( )

4( ) ( ) (30)

kk k

k k

0 0

2

and the subsidiary condition gives

ρρ

=

=

td t

dtien k f t

( )( )

( ) . (31)

kk

k0

Using this condition we can rewrite the expression of theLagrangian in the form containing only ρ ρ *t t( ) , ( )

k kand

ρ ρ *t t( ) , ( )k k

:

⎡⎣ ⎤⎦∑ ρ ρ ω ρ ρ

=

× * − *

L te

m

n

kt t t t

( )1

21

( ) ( ) ( ) ( ) . (32)pl

kk k k k

0 20

22

Setting

ρ

ρ

= +

* = −

e

m

n kt q t iq t

e

m

n kt q t iq t

1 1( ) ( ) ( ) ,

1 1( ) ( ) ( ) ,

(33)

( ) ( )

( ) ( )

k k k

k k k

0

1 2

0

1 2

q t( )( )ik

with i= 1, 2 being the real functions of t, we obtain the

formula of the Lagrangian of a system of harmonic oscillators

with the generalized real coordinates q t( )( )ik

and the fre-

quency ωpl:

⎡⎣ ⎤⎦∑ ∑ ω= −=

L t q t q t( )1

2( ) ( ) . (34)( ) ( )

i

ipl

i

kk k0

1,2

2 2 2

In order to quantize the system with the Lagrangian (34)

we must calculate the generalized momenta p t( )( )ik

as well as

the Hamiltonian H. We obtain

=∂∂

= p tL t

q tq t( )

( )

( )( )( )

( )( )i

i

ik

k

k0

and

⎡⎣ ⎤⎦∑ ∑ ω= +=

H p t q t1

2( ) ( ) . (35)( ) ( )

i

ipl

i

kk k

1,2

2 2 2

The quantization procedure consists of the replacement

of dynamical canonical variables q t( )( )ik

and p t( )( )ik

by the

corresponding hermitian operators q ( )ik

and p( )ik, i= 1, 2,

satisfying the following canonical commutation relations

⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦⎡⎣ ⎤⎦ δ δ

ˆ ˆ − ˆ ˆ =

ˆ ˆ = − ℏ

p p q q

p q i

, , 0,

, . (36)

i j i j

i jij

k( )

l( )

k( )

l( )

k( )

l( )

kl

Hamiltonian of the quantized system is the operator

⎡⎣⎢

⎤⎦⎥∑ ∑ ω= ˆ + ˆ

=

H p q1

2. (37)( ) ( )

i

ipl

i

kk k

1,2

22 2

Instead of two hermitian operators q ( )ik

and p( )ik

for each

set of two indices i and k we introduce two operators

ω

ω

ω

ω

ˆ =ℏ

ˆ −ℏ

ˆ

ˆ =ℏ

ˆ +ℏ

ˆ+

a p i q

a p i q

1

2 2,

1

2 2. (38)

( ) ( ) ( )

( ) ( ) ( )

i

pl

i pl i

i

pl

i pl i

k k k

k k k

They satisfy commutation relations

⎡⎣ ⎤⎦⎡⎣ ⎤⎦ δ δ

ˆ ˆ = ˆ ˆ =

ˆ ˆ =

+ +

+

a a a a

a a

[ , ] , 0,

, . (39)

( ) ( ) ( ) ( )

( ) ( )

i j i j

i jij

k l k l

k l kl

Inversely we can express p( )ik

and q ( )ik

in terms of a ( )ik and

ˆ+

a ( )ik

⎡⎣ ⎤⎦⎡⎣ ⎤⎦

ω

ω

ˆ =ℏ

ˆ + ˆ

ˆ = ℏ ˆ − ˆ

+

+

p a a

q i a a

2,

2, (40)

( ) ( ) ( )

( ) ( ) ( )

i pl i i

i

pl

i i

k k k

k k k

and obtain the Hamiltonian operator in the form

⎜ ⎟⎛⎝

⎞⎠∑ ∑ω= ℏ ˆ ˆ +

=

+H a a

1

2. (41)( ) ( )

pl

i

i i

kk k

1,2

Operators a ( )ik and ˆ

+a ( )i

k satisfying the commutationrelations (39) play the role of the destruction operator andcreation operator, respectively, for a quasiparticle of the type iand with the momentum ℏk. The quasiparticles of this kindare the quanta of the collective excitations, i.e. of the fluc-tuations of the electron number density n tr( , ) in the electrongas. They are called plasmons.

Adv. Nat. Sci.: Nanosci. Nanotechnol. 5 (2014) 025001 V H Nguyen and B H Nguyen

4

Since the electron gas behaves like a system of harmonicoscillators in the above presented lowest order approximationwith respect to the small oscillating displacements of elec-trons, this approximation in often called the harmonicapproximation. Note that in the harmonic approximation theplasma energy does not depend on its momentum and equals

ωℏ pl. The plasmons in the system with Lagrangian L0(t) do

not participate in any interaction process and are called thefree plasmons.

4. Beyond harmonic approximation andplasmon–plasmon interaction

Now we return to the total Lagrangian (9) and the subsidiarycondition (16) and study the fluctuations of the charge densityρ tr( , ) beyond the harmonic approximation. Substituting theFourier series (19) and (29) for ρ tr( , ) and δ tr r( , ) into bothsides of the continuity equation (16), we obtain followingsystem of equations between their Fourier components

∑

ρ

ρ

= −

× −

t ien k f ti

V

tl

f tkl

( ) ( )

( ) ( ) . (42)

k k

lk l l

0

In order to express f t( )k

in term of ρ t( )k

and ρ t( )k

we

rewrite equation (42) in another form

∑

ρ

ρ

= − +

× −

f ti

en kt

en V

kt

lf tkl

( )1

( )1 1

1( )

1( ) (43)

k l

k k

ll

0 0

and solve the system of equations (43) with respect to f t( )k

by means of the iteration procedure. Then we obtain the

expression of f t( )k

in the form of a functional power series of

ρ t( )k

and ρ t( )k

. Up to the third order we have

∑

∑

∑

ρ

ρ ρ

ρ

ρ ρ

= − −

× −

×

× ′′

+

−

−

′− ′ ′

f ti

en kt

i

en k V

kt

lt

i

en k V

kt

l V

tl V

t

kl

kl

ll

( )1

( )( )

1 1

1( ) ( )

( )

1 1

1( )

1

( )1

( )

higher order terms. (44)

k k

lk l l

lk l

ll l l

0 02

20

3

2

2

Consider the total Lagrangian (9). Using the decom-positions (19) and (29) for ρ tr( , ) and δ tr r( , ) , we expressL(t) in terms of their Fourier components ρ t( )

kand f t( )

kas

follows

⎡⎣⎤⎦⎥

⎡⎣⎤⎦

∑

∑ ∑

π ρ ρ

ρ

ρ

= *

− *

−

× *

+ * *

+

+

L t mn f t f t

kt t

m

e V

klt f t f t

f t f t t

kl

( )1

2( ) ( )

4( ) ( )

1

4

1

( ) ( ) ( )

( ) ( ) ( ) . (45)

kk k

k k

k lk l k l

k l k l

0

2

Substituting the expansion (44) of f t( )k

into the r.h.s. of

formula (45) and dividing the derived expression of the totalLagrangian L(t) into two parts

= +L t L t L t( ) ( ) ( ) , (46)0 int

where L0(t) is the expression (32) of the Lagrangian in theharmonic approximation and Lint(t) is called the interactionLagrangian. In the fourth order with respect to the functionsρ t( )

k, ρ *t( )

kand ρ t( )

k, ρ *t( )

kwe obtain

⎡⎣⎢

⎤⎦

∑ ∑

∑ ∑ ∑

ρ ρ ρ

ρ ρ ρ ρ

=

*

+′

′

× * *

−

′

− ′ ′ −

L tmn

en

V k lt t t

en V k l l

t t t t

kl

kl ll

( )1

2 ( )

1( ) ( ) ( )

1 1 ( ) ( )

( ) ( ) ( ) ( ) . (47)

k lk k l l

k l l

l l l k l k

int0

03

2 2

02 2 2

Note that due to the property (24) of the functions ρ t( )k

and

ρ t( )k

, interaction Lagrangian (47) is a real function of t.

5. Conclusion and discussion

In this work we have elaborated the quantum theory of col-lective excitations of the isotropic and homogeneous electrongas in the framework of the canonical formalism. The quantaof these collective excitations are the quasiparticles of somespecial type called plasmons. In this formalism there naturallyappear the destruction and creation operators of plasmonsduring the standard quantization procedure. By applying thismethod we can establish the Lagrangian of the interactionbetween plasmons and other particles or quasiparticles incondensed matters from the first principle.

In the harmonic approximation the Lagrangian of thesystem of electrons in the electron gas can be represented asthat of the free plasmons with a common momentum-inde-pendent energy ωℏ pl. In higher order approximation with

respect to the collective excitations of the electron gas, thetotal Lagrangian consists of two parts, one is that of freeplasmons, another is the interaction Lagrangian describing theplasmon–plasmon interaction. The dependence of the energyof a plasmon on its momentum is the consequence of the

Adv. Nat. Sci.: Nanosci. Nanotechnol. 5 (2014) 025001 V H Nguyen and B H Nguyen

5

renormalization of the one-plasmon state vector (or two-pointGreen function of the plasmon) due to the plasmon–plasmoninteraction.

As a simple example of the interaction of plasmons withother particles let us consider the photon–plasmon interaction.The photons are described by a quantized elecromagnetic

field with the vector potential ˆ tA r( , ) and the scalar potential

V tr( , ) . The Lagrangian of the interaction between photonsand all electrons of the electron gas is

∫

∫∫

δ ˆ= ×

+

+ − ˆ

γ−L t me

cd n t t t

me

cd n t t

e d n t n V t

r r r r A r

r r A r

r r r

( ) ( , ) ( , ) ( , )

1

2( , ) ( , )

[ ( , ) ] ( , ) . (48)

e

2

22

0

where c is light velocity in the vacuum.In terms of the displacements δ tr r( , ) and the charge

density ρ tr( , ) we have

∫∫

∫∫

∫

δ

ρ δ

ρ

ρ

ˆ

ˆ

ˆ

=

−

+

−

− ˆ

γ−L t mne

cd t t

m

cd t t t

mne

cd t

me

cd t t

d t V t

r r r A r

r r r r A r

r A r

r r A r

r r r

( ) ( , ) ( , )

( , ) ( , ) ( , )

1

2( , )

1

2( , ) ( , )

( , ) ( , ) . (49)

e 0

0

2

22

22

By canonically quantizing the dynamical variableδ tr r( , ) and ρ tr( , ) , we obtain the plasmon–photon inter-action Lagrangian.

Acknowledgment

The authors would like to express their gratitude to theVietnam Academy of Science and technology for the supportto this work.

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Adv. Nat. Sci.: Nanosci. Nanotechnol. 5 (2014) 025001 V H Nguyen and B H Nguyen

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